1. Field of Use
This invention relates generally to transformers. More specifically, this invention concerns transformers which have low leakage reactance.
2. Description of the Prior Art
a. Conventional Transformer Structures PA1 b. Undesirable Load-Induced Harmonics
Leakage reactance, L.sub.l, is present in all transformers and is caused by flux set up by the primary coil that does not link the secondary coil or vise versa. Primary or secondary winding leakage flux returns upon itself either through the core or by looping around in the gap between the primary and secondary coils. As a consequence, it is an important objective in transformer design to ensure that the primary coil is surrounded to the fullest extent possible by the secondary coil. Leakage reactance is detrimental for a number of reasons including that it decreases voltage regulation and decreases the dynamic response, especially for high frequency AC based systems.
Leakage reactance is a function of coil dimensions, winding-to-winding spacing, and the amount of primary to secondary coil area that is overlapped. Leakage reactance, L.sub.l, is defined by Eq. 1, the parameters of which are illustrated in the conventional interleaved transformer construction of FIG. 1: ##EQU1## where L.sub.l is the leakage inductance of both windings in Henry's; N is the number of turns; MT is the mean length of a turn in inches for a whole coil; n is the number of dielectrics between windings; c is the thickness in inches of dielectric between windings; a is the winding height in inches; and b is the winding width in inches. As is seen from Eq. 1, leakage reactance is reduced by using a small number of turns, short mean turn, and low, wide core windows. Eq. 1 assumes symmetrical coverage of primary-secondary windings. Leakage inductance may increase by a factor of 20 for a secondary coil layer that covers only 10% of the exposed primary coil area.
Leakage reactance is reduced by interleaving in conventional transformers. Leakage reactance in conventional transformers can be reduced but not eliminated because it is difficult to prevent leakage flux between the interleaved spaces c. Moreover, leakage reactance in conventional transformer structures is strongly coupled to the magnitude of the mutual or magnetizing inductance L.sub.m as seen in the transformer per phase equivalent circuit of FIG. 2 and Eq. 2. The parameter k is the primary-secondary coupling coefficient and is less than 1.0. ##EQU2## Magnetizing inductance L.sub.m is the inductance needed to magnetize the core and, unlike leakage reactance, is desired to be as high as possible. Thus, L.sub.l and L.sub.m represent a design trade-off. As L.sub.m is increased by, for instance, increasing the number of turns, L.sub.l increases by the number of turns squared or N.sup.2. A second trade-off exists between L.sub.e and L.sub.m in high voltage transformer applications. As the primary-secondary dielectric voltage stress is increased, the required dielectric insulation spacing between coils, c, also increases from Eq. 1. This tends to increase L.sub.l because the larger spacing between coils provides a larger leakage path between primary and secondary coils.
One unsatisfactory solution to the leakage reactance problem in transformers is the coaxial transformer, a portion of which is shown in FIG. 3. As is seen there, any leakage from the inner primary coil must pass through the secondary coil thereby ensuring that the leakage is linked through the secondary coil. The coaxial structure reduces L.sub.l because inductance is determined by the distance between the inner and outer coaxial coils and the internal self inductance of the conductors. The inductance for the 1:1 turns ratio coaxial structure of FIG. 4 is determined by Eq. 3. ##EQU3## The "1/4" first term of Eq. 3 is due to the internal self inductance of a round cylindrical conductor. The second term, In R.sub.2 /R.sub.1, represents leakage inductance resulting from the geometric spacing of the primary and secondary conductors. The third term may be considered an internal self inductance of the thin hollow tube secondary conductor. The third term is typically small in value because the R.sub.2 and R.sub.3 radius values are close in magnitude as can be seen from FIG. 4. The outer surface area of R.sub.3 dissipates conductor heat and allows the equivalent R.sub.2 -R.sub.3 conductor area to be reduced to a thin hollow tube.
Minimal leakage inductance occurs when the outer conductor is placed directly over the inner conductor and the intersticial space required for insulation, creapage, and clearance is minimized. This drives the ln(R.sub.2 /R.sub.1) term towards zero. In the configuration in FIG. 4, the inner conductor internal self inductance "1/4" term dominates the effective inductance regardless of the R.sub.1 and R.sub.2 radius values. For example, the arithmetic sum of the first and second term of Eq. 3 result in a 0.253992 value with R.sub.1 =0.5 inches and R.sub.2 =0.502 inches, while a sum of 0.250666 results with R.sub.1 =3.0 and R.sub.2 =3.002 inches, a total difference of 1%. Thus, the dominant internal self inductance "1/4" term of Eq. 3 is essentially constant for solid cylindrical conductors. Skin and proximity effects of the ln(R.sub.2 /R.sub.1) external inductance term and of the secondary hollow tube will have an immeasurable effect due to the dominant first term. Thus, conventional coaxial transformer structures have limited capacity to substantially reduce leakage reactance.
Furthermore, when the coaxial structure is designed to optimize the L.sub.m and L.sub.l trade-off, the transformer tends to be very long and requires many toroidal cores stacked together. As core length and the number of cores increases, the transformer becomes unwieldy and unattractive for most applications and manufacturing costs tend to rise dramatically.
Referring again to FIG. 2 undesirable higher order harmonics such as the fifth, seventh, eleventh, thirteenth, fifteenth, seventeenth harmonics, and so on are caused by a number of different factors in a typical transformer application. Such nonlinear harmonics can be caused by the load itself, such as a solid state converter, or extraneous sources such as fluorescent lighting. These harmonics generate a harmonic current I.sub.H, shown in FIG. 2, which are injected back toward the transformer and the utility from the load. If I.sub.H is large enough, devices attached up-line may experience problems due to these downline nonlinear harmonics.
FIG. 5A depicts a typical desired sine wave to be transmitted to the load from the utility. FIG. 5B depicts that same signal with undesirable harmonics in it. The problems of load-induced higher order harmonics has been brought into focus as of late as a result of new IEEE standards regarding harmonics. These standards are now better defined and likely to be enforced. Power users who generate substantial harmonics which are detrimental to up-line users, including the utility itself, will likely be liable for disruptions.
FIG. 6 depicts four different load current waveforms I.sub.line containing higher order harmonics that are commonly induced by loads such as solid-state converters. As is seen, FIG. 6A is a six pulse converter with DC link L-C filter under light load conditions, FIG. 6B is a six pulse converter with DC link L-C filter under lull load conditions, FIG. 6C is a twelve pulse converter with large DC link inductor, and FIG. 6D is a six pulse converter with large DC link inductor. As is seen, some of the currents generated are discontinuous and may have high peaks compared to RMS values. It is desirable to cancel out the higher order harmonics to recapture the fundamental component of the current so that up-line disruptions are eliminated.
In a typical transformer application, higher order harmonics as high as the seventeenth harmonic should be suppressed. Beyond the seventeenth harmonic, inductance at the utility presents a sufficiently high impedance which masks the effect of the undesirable harmonics. Assuming that harmonics up to the seventeenth harmonic should be suppressed, it can be seen from Eq. 4 that the minimum frequency, f.sub.min, that is required to cancel out the highest order harmonic number and return the signal to its fundamental order (e.g., fundamental =60 Hz) is: EQU f.sub.min =f.sub.fundamental .multidot.n.sub.harmonic .multidot.n.sub.encoder. Eq. 4
The encoding number from Eq. 4 acknowledges that in order to use a pulse width modulation system (PWM), in the compensator, one must use an encoding frequency that is twelve times greater than the frequency of the highest harmonic. The actual switching frequency of the device in the compensator is at the f.sub.min frequency.
Thus, substituting 17 for n.sub.harmonic in Eq. 4 it can be seen that a 12 kHz signal must be injected into the transformer primary to compensate for the seventeenth harmonic. Injection of such a high frequency, high current component into a conventional transformer, however. will create a large voltage drop across inductance L.sub.l in FIG. 2 according to Eq. 5: EQU V.sub.l =I.sub.line .multidot.X.sub.l .multidot. Eq. 5
Where V.sub.l is the voltage drop across L.sub.l, L.sub.line is the injected current, and X.sub.l is the leakage reactance of the transformer. Eq. 5 can be rewritten as follows: EQU V.sub.l =I.sub.line 2.pi.f.sub.min L.sub.l .multidot. Eq. 6
As the voltage across inductance L.sub.l increases, voltage stress increases which increases the necessary volt-ampere rating and required size of the components. This is undesirable for obvious reasons.
The second problem encountered when attempting to inject a signal of such high frequency and high current into a conventional transformer having significant leakage reactance is illustrated by Eq. 7: ##EQU4## where V.sub.comp is the voltage in the compensating device, V.sub.load is the voltage at the load, and di/dt is the rate of change of current. It can be seen from Eq. 7 that if leakage reactance is high, di/dt is too low, thus making it very difficult if not impossible to compensate for the higher order harmonics in the signal; the dynamic response of the signal injector cannot be made high enough to follow the rapidly changing rate of current.
Further complicating matters is that as the minimum compensating frequency f.sub.min is increased, X.sub.l increases which requires higher voltage compensation and greater dynamic response. The dynamic response of any compensator in the system is limited because of the fixed volt-ampere (VA) size of the compensator.
Referring again to Eq. 2 illustrating the close coupling between L.sub.l and L.sub.m, it can be seen that if k=0.9995, which is the best known k available for conventional transformers, and L.sub.m is chosen to meet a predetermined VA size of the compensator, the L.sub.l solved for typically would be much too high to allow sufficient compensation for undesirable harmonics.